The sixteenth century saw the final acceptance of negative numbers, integral and
fractional. The seventeenth century saw decimal fractions with the modern notation
quite generally used by mathematicians. The next hundred years saw the imaginary
become a powerful tool in the hands of De Moivre, and especially of Euler. For the
nineteenth century it remained to complete the theory of complex numbers, to separate
irrationals into algebraic and transcendent, to prove the existence of transcendent
numbers, and to make a scientific study of a subject which had remained almost
dormant since Euclid, the theory of irrationals. The year 1872 saw the publication
of the theories of Weierstrass (by his pupil Kossak), Heine (Crelle, 74), G. Cantor
(Annalen, 5), and Dedekind. Méray had taken in 1869 the same point of departure as
Heine, but the theory is generally referred to the year 1872. Weierstrass’s method has
been completely set forth by Pincherle (1880), and Dedekind’s has received additional
prominence through the author’s later work (1888) and the recent indorsement by
Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series,
while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers,
separating all rational numbers into two groups having certain characteristic properties.
The subject has received later contributions at the hands of Weierstrass, Kronecker
(Crelle, 101), and Méray.
Continued Fractions, closely related to irrational numbers and due to Cataldi,
1613),1 received attention at the hands of Euler, and at the opening of the nineteenth
century were brought into prominence through the writings of Lagrange. Other
noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857),
Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with
determinants, resulting, with the subsequent contributions of Heine, Möbius, and
Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general
theory, as have numerous contributors to the applications of the subject.
Transcendent Numbers2 were first distinguished from algebraic irrationals by
Kronecker. Lambert proved (1761) that π cannot be rational, and that en (n being
a rational number) is irrational, a proof, however, which left much to be desired.
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History of Modern Mathematics ~ Irrational and Transcendent Numbers
Legendre (1794) completed Lambert’s proof, and showed that π is not the square root
of a rational number. Liouville (1840) showed that neither e nor e2 can be a root of
an integral quadratic equation. But the existence of transcendent numbers was first
established by Liouville (1844, 1851), the proof being subsequently displaced by G.
Cantor’s (1873). Hermite (1873) first proved e transcendent, and Lindemann (1882),
starting from Hermite’s conclusions, showed the same for π. Lindemann’s proof was
much simplified by Weierstrass (1885), still further by Hilbert (1893), and has finally
been made elementary by Hurwitz and Gordan.
1 But see Favaro, A., Notizie storiche sulle frazioni continue dal secolo
decimoterzo al deci mosettimo, Boncompagni’s Bulletino, Vol. VII, 1874, pp. 451, 533.
2 Klein, F., Vorträge über ausgewählte Fragen der Elementargeometrie, 1895, p.
38; Bachmann, P., Vorlesungen über die Natur der Irrationalzahlen, 1892.
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